10³ times 8 bacteria will Harriet have in the dish at the end of 6 hours, if she has 10³ bacteria now and they double every 2 hours, option D.
Starting with the initial number of bacteria: 10³
Since the bacteria double every 2 hours, after 2 hours, there will be 10³ × 2 bacteria.
After another 2 hours (total of 4 hours), the bacteria will double again: (10³ × 2) × 2 = 1³ × 2²
After the final 2 hours (total of 6 hours), the bacteria will double once more: (10³ × 2²) × 2 = 10³ × 2³
So, at the end of 6 hours, Harriet will have 10³ × 2³ bacteria in the dish. The correct answer is D. 10³ times 8.
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A system of linear equations is shown on the graph.
The graph shows a line that passes through negative 10 comma 10, negative 5 comma 9, and 0 comma 8. The graph also shows another line that passes through negative 8 comma 12, negative 5 comma 9, and 0 comma 4.
What is the solution to the system of equations?
There are infinitely many solutions.
There is no solution.
There is one unique solution (−5, 9).
There is one unique solution (0, 8).
The correct statement regarding the solution to the system of equations is given as follows:
There is one unique solution (−5, 9).
How to define a linear function?The slope-intercept representation of a linear function is given by the equation shown as follows:
y = mx + b
The coefficients m and b have the meaning presented as follows:
m is the slope of the function, representing the increase/decrease in the output variable y when the input variable x is increased by one.b is the y-intercept of the function, representing the numeric value of the function when the input variable x has a value of 0. On a graph, it is the value of y when the graph of the function crosses or touches the y-axis.For the first line, the points are given as follows:
(-10, 10) and (0,8).
Hence the equation is:
y = -0.2x + 8.
For the second line, the points are given as follows:
(-8, 12) and (0,4).
Hence the equation is:
y = -x + 4.
Then the x-coordinate of the solution is obtained as follows:
-0.2x + 8 = -x + 4
0.8x = -4
x = -5.
The y-coordinate is given as follows:
y = -(-5) + 4 = 9.
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Identify the equation of the line that passes through the pair of points (−3, 6) and (−5, 9) in slope-intercept form.
Therefore, the equation of the line that passes through the points (-3, 6) and (-5, 9) in slope-intercept form is:
y = -3/2 x + 3/2
Step-by-step explanation:
To find the equation of the line that passes through the points (-3, 6) and (-5, 9) in slope-intercept form (y = mx + b), we need to first find the slope of the line using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) = (-3, 6) and (x2, y2) = (-5, 9).
m = (9 - 6) / (-5 - (-3))
m = 3 / (-2)
m = -3/2
Now that we have the slope, we can use either of the two given points and the slope to find the y-intercept (b) of the line:
y = mx + b
6 = (-3/2)(-3) + b
6 = 9/2 + b
b = 6 - 9/2
b = 3/2
Solve for the unknown value.
x
61
42
degrees.
Hi! To solve for the unknown value in the equation "x + 61 + 42 = degrees", follow these steps:
1. Combine the known values (61 and 42) by adding them together: 61 + 42 = 103.
2. Rewrite the equation with the combined values: x + 103 = degrees.
3. To isolate the unknown value (x), subtract 103 from both sides of the equation: x = degrees - 103.
So, the unknown value x can be found by subtracting 103 from the given degrees value.
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find the missing side length
Answer:
Step-by-step explanation:
[tex]x^{2}[/tex]=a²+b²
x²=6²+2.5²
x²=36+6.25
x²=42.25
√x=√42.25
x=6.5
Answer:
6.5
Step-by-step explanation:
6^2 + 2.5^2 = c^2
•36 + 6.25 = c^2
•42.25 = c^2
• find square root of 42.25 = 6.5
• 6.5 = C
What is the shape of the height and weight distribution? A. The height and weight distribution exhibit a negative and a positive skew, respectively. B. Both the height and weight distribution exhibit a positive skew. C. Both the height and weight distribution exhibit a negative skew. D. Both the height and weight distribution are symmetric about the mean. E. The height and weight distribution exhibit a positive and a negative skew, respectively
D. Both the height and weight distribution are symmetric about the mean.
What is the shape of the height and weight distribution? If a distribution is symmetric about the mean, it means that the values are evenly distributed on either side of the mean, resulting in a bell-shaped curve. The height and weight of individuals in a population tend to follow this type of distribution, with the majority of individuals clustering around the mean height and weight values. This is known as a normal distribution, which is a type of symmetric distribution. Therefore, option D is the correct answer. Options A, B, C, and E are not correct because they indicate skewness in the distribution, which is not typically observed in height and weight data.Learn more about distribution,
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The library in natchitoches had about 60000 volumes in march 2004 and was adding about 400 volumes per month Write an equation expressing y, the total number of volumes in the library, in terms of x, the number of mouths after march 2004
The equation expressing y, as the total number of volumes in the library, in terms of x, the number of mouths after march 2004 is y = 400x + 60,000.
What is the equation for the volumes in library?The equation expressing y, as the total number of volumes in the library, in terms of x, the number of mouths after march 2004 is determined as follows;
Using general linear equation;
y = mx + b
where:
y is the total number of volumes in the libraryx is number of months after March 2004m is the rate of change b is the initial number of volumes in the library in March 2004The initial volume as at March = 60,000
the rate = 400 volumes / month
The equation is determined as;
y = 400x + 60,000
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Helps me please What is most likely to push the price of a company's stock higher?
O A. An increase in demand for the company's stock
OB. An increase in demand for the company's products
O C. An increase in tariffs paid by the company's competitors
O D. An increase in the exchange rate for the US dollar
SUBMIT
PREVIOUS
e to search
O
RI
The most likely factor to push the price of a company's stock higher is an increase in demand for the company's stock. Option A.
A stock's price will often increase when there is greater demand than there is supply. Various things, such as good news about the company's financial performance, new product releases or innovations, positive analyst reports, or general market trends, can contribute to this increased demand.
The company's financial performance may benefit from a rise in product demand, but if investors do not view it as a key growth engine for the business, it may not necessarily transfer into a higher stock price.
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if Ashely sold 98 cups for $1 each, how much profit did she make
Answer:
After selling 98 cups, she made $98
Step-by-step explanation:
If she sold 98 cups, and each one went for $1, then you can set up an equation:
98(1)=98
Answer:
Ashley made $98.
Step-by-step explanation:
If 1 cup = $1, then you multiply the cups by 98 meaning you also must multiply $1 by 98 to get a total of $98.
Help!!!
if abc is similar to xyz and yzx, what special type of triangle is abc? complete the explanation.
If ABC is similar to XYZ and YZX, then the corresponding angles of those triangles are same, and the corresponding sides are proportional. because of this triangle ABC is a special type of triangle known as a "similar triangle."
In a similar triangle, the angles of the triangle are equal, however the sides can be exclusive lengths. but, the ratios of the corresponding aspects are usually the same. This property is beneficial in lots of regions of mathematics and physics, which includes trigonometry and the study of geometric shapes.
inside the case of triangle ABC, the fact that it's far much like each XYZ and YZX tells us that its angles are same to those of these triangles, and its sides are proportional to the ones of these triangles. This property may be used to solve many issues regarding triangles and other geometric shape.
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Find the volume of a cylinder with a diameter of 28 meters and a height of 6 and one half meters. Approximate using pi equals 22 over 7.
28,028 cubic meters
4,004 cubic meters
1,274 cubic meters
572 cubic meters
The volume of the cylinder is 4004 cubic metres.
How to find the volume of a cylinder?The diameter of the cylinder is 28 metres and the height of the cylinder is 6.5 metres.
Therefore, the volume of the cylinder can be found as follows:
Hence,
volume of a cylinder = πr²h
where
r = radiush = heightTherefore,
volume of the cylinder = 22 / 7 × 14² × 6.5
volume of the cylinder = 22 / 7 × 196 × 6.5
volume of the cylinder = 28028 / 7
volume of the cylinder = 4004 cubic metres
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Find and interpret the mean absolute deviation of the data. 46,54,43,57,50,62,78,42
In this case, the mean absolute deviation of 8.5 indicates that, on average, the data points deviate from the mean by about 8.5 units.
What is mean absolute deviation?Mean absolute deviation (MAD) is a statistical measure that represents the average distance between each data point and the mean of the data set. It is calculated by finding the absolute value of the difference between each data point and the mean, and then taking the average of these absolute differences. MAD is a useful measure of the variability or spread of a data set, and is often used as an alternative to the more common measure of standard deviation. Like standard deviation, MAD gives an indication of how spread out the data is, but unlike standard deviation, MAD is less sensitive to extreme values or outliers.
Here,
To find the mean absolute deviation of the data, we first need to calculate the mean (average) of the data:
Mean = (46 + 54 + 43 + 57 + 50 + 62 + 78 + 42) / 8
Mean = 52
The mean of the data is 52.
Next, we need to calculate the absolute deviation of each data point from the mean. The absolute deviation is simply the absolute value of the difference between each data point and the mean:
|46 - 52| = 6
|54 - 52| = 2
|43 - 52| = 9
|57 - 52| = 5
|50 - 52| = 2
|62 - 52| = 10
|78 - 52| = 26
|42 - 52| = 10
Now, we can calculate the mean absolute deviation by taking the average of the absolute deviations:
Mean Absolute Deviation = (6 + 2 + 9 + 5 + 2 + 10 + 26 + 10) / 8
Mean Absolute Deviation = 8.5
The mean absolute deviation of the data is 8.5.
Interpretation: The mean absolute deviation represents the average distance between each data point and the mean of the data. In this case, the mean absolute deviation of 8.5 indicates that, on average, the data points deviate from the mean by about 8.5 units. This means that the data points are relatively spread out, with some points being much higher or lower than the mean.
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Determine how long it will take for 650 mg of a sample of chromium-51, which has a half life of 28 days, to decay to 200 mg.
It will take approximately 60.9 days for 650 mg of chromium-51 to decay to 200 mg.
What is Equation ?
An equation is a mathematical statement that shows that two expressions are equal. It typically contains variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.
The decay of a radioactive substance can be modeled by the following equation:
N(t) = N₀ * [tex](1/2)^{(t/T) }[/tex]
where:
N(t) is the amount of the substance remaining after time t
N₀ is the initial amount of the substance
T is the half-life of the substance
We can use this equation to find how long it will take for 650 mg of chromium-51 to decay to 200 mg.
Let's first find the decay constant (λ) for chromium-51:
λ = ㏒(2) ÷ T = ㏒(2) ÷ 28 = 0.0248 (rounded to 4 decimal places)
Now we can use the equation:
N(t) = N₀ * [tex]e^{(-λ*t)}[/tex]
We know that N₀ = 650 mg and N(t) = 200 mg, so we can solve for t:
200 = 650 * [tex]e^{(-0.0248*t)}[/tex]
Dividing both sides by 650:
0.3077 = [tex]e^{(-0.0248*t)}[/tex]
Taking the natural logarithm of both sides:
㏒(0.3077) = -0.0248*t
Solving for t:
t = ㏒(0.3077) : (-0.0248) ≈ 60.9 days
Therefore, it will take approximately 60.9 days for 650 mg of chromium-51 to decay to 200 mg.
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1 ) If the supplement of an angle is 30 degrees more than the measure of the angle, what is the measure of the angle?
2) If the supplement of an angle is 12 degrees less than twice the measure of the angle, what is the measure of the angle?
a) If the supplement of an angle is 30 degrees more than the measure of the angle, the measure of the angle is 75 degrees.
b) If the supplement of an angle is 12 degrees less than twice the measure of the angle, the measure of the angle is 64 degrees.
a) Let x be the measure of the angle. The supplement of the angle is 180-x. According to the problem, the supplement of the angle is 30 degrees more than the measure of the angle. This can be written as:
180 - x = x + 30
Solving for x, we get:
2x = 150
x = 75
b) Let x be the measure of the angle. The supplement of the angle is 180-x. According to the problem, the supplement of the angle is 12 degrees less than twice the measure of the angle. This can be written as:
180 - x = 2x - 12
Solving for x, we get:
3x = 192
x = 64
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Given m||n, find the value of x
Answer:
x=32°
Step-by-step explanation:
3x-2=2x+30
x-2=30
x=32°
The ratio of an objects weight on earth to its weight on the moon is 6:1 the first person to walk on the moon was neil armstrong. he weighed 165 pounds on earth. what would be the proportion of this word problem?
The proportion of this word problem is 6 : 1 where Neil Armstrong weighed approximately 27.5 pounds on the moon.
The proportion of a word problem represents the relationship between two or more quantities. In this case, the proportion can be set up as:
Weight on Earth : Weight on Moon = 6 : 1
Using the information provided in the problem, we know that Neil Armstrong weighed 165 pounds on Earth. We can use this information to find his weight on the moon by setting up a proportion:
165 : x = 6 : 1
where x represents his weight on the moon. To solve for x, we can cross-multiply and simplify:
165 * 1 = 6 * x
x = 165/6
x ≈ 27.5
Therefore, Neil Armstrong weighed approximately 27.5 pounds on the moon.
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from a circular sheet of a radius 5cm, a circle of radius 3cm is removed. find the area of the remaining sheet
Check the picture below.
[tex]\textit{area of a circular ring}\\\\ A=\pi (R^2 - r^2) ~~ \begin{cases} R=\stackrel{outer}{radius}\\ r=\stackrel{inner}{radius}\\[-0.5em] \hrulefill\\ R=5\\ r=3 \end{cases}\implies A=\pi (5^2-3^2)\implies A\approx 50.27~cm^2[/tex]
How long does it take time a car to move 150 centimeters? we all have a good idea that if it takes a smaller period of time to move 50 centimeters? the car must be moving faster. But how do we measure the space of a toy car in your own car, its is easy because engineers have designed a speedometer that automatically tells you how fast your car is moving.
If you record the time it takes for the car to move 50 centimeters and then record the time it takes to move 150 centimeter, will there be a difference in the speed of the car between the tow runs? Explain your prediction
There may be a difference on speed based on various factors
How to determine speedThough it would be reasonable to conclude that if it takes less time for a car to move 50 centimeters than 150 centimeters, they must be traveling faster; we do not know exactly when they did this and whether their speed changed accordingly.
Furthermore, it could have taken the same time covering both distances; thus leaving unanswered questions regarding their relative speeds.
Due to variables affecting car speed such as changes in terrain or obstacles in its path, we cannot accurately predict its speed without more information regarding these two runs.
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1 A) (In(x) +1) 2x In(2) Denivate h(x) = √xen(x) h( 1 B) In() + V2V C) V 2. In(x) + 2 D) In (30) + 2ln()
The derivatives of the given functions are:
A) h'(x) = (1/2)√xen(x)[2+In(x)]
B) h'(x) = (1/x) - V2V
C) h'(x) = (2/x) + 2
D) h'(x) = 0
A) To find the derivative of h(x) = √xen(x), we use the product rule of differentiation. Let u = √x and v = en(x).
Then, h(x) = uv, and h'(x) = u'v + uv'.
We have u' = (1/2)x^(-1/2) and v' = en(x)(1/x).
Substituting the values, we get h'(x) = (1/2)√xen(x)[2+In(x)].
B) To find the derivative of h(x) = In(x) + V2V, we use the sum rule of differentiation.
Using the properties of logarithms, we rewrite the function as h(x) = In(x) + (1/2)ln(x).
Taking the derivative, we get h'(x) = (1/x) - V2V.
C) To find the derivative of h(x) = V2 In(x) + 2, we use the sum rule of differentiation.
Taking the derivative, we get h'(x) = (2/x) + 2.
D) To find the derivative of h(x) = In(30) + 2ln(x), we use the sum rule of differentiation.
Taking the derivative, we get h'(x) = 0, since the derivative of a constant is always zero.
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John's guest house is 18 m long, and 15 m wide. What is the length of the diagonal of the house?
The length of the diagonal of the house is 23.43 meters based on the dimensions of the John's guest house.
The diagonal of the house will be calculated using Pythagoras theorem. We know that diagonal will be hypotenuse and length and width will be base and perpendicular.
Diagonal = ✓length² + width²
Keep the values in formula
Diagonal = ✓18² + 15²
Taking square
Diagonal = ✓324 + 225
Adding the values
Diagonal = ✓549
Taking square root
Diagonal = 23.43 meters
Hence, the diagonal of the house is 23.43 meters.
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Match the formulas for volume and calculate the volumes of the sphere, cylinder, and cone shown below. Each shape has a radius of 2.5 and the cylinder and cone have a height of 4.
options for each drop down box [choose]:
Sphere - volume measure
Sphere - volume formula
Cone - volume formula
Cone - volume measure
Cylinder - volume measure
Cylinder - volume formula
None of these options
Answer:
The formula for the volume of a cone is ⅓ r2h cubic units, where r is the radius of the circular base and h is the height of the cone.The volume of any sphere is 2/3rd of the volume of any cylinder with equivalent radius and height equal to the diameter.The formula for the volume of a sphere is 4⁄3πr³. For a cylinder, the formula is πr²h. A cone is ⅓ the volume of a cylinder, or 1⁄3πr²h
Step-by-step explanation:
The formula for volume is: Volume = length x width x height
Answer:
Step-by-step explanation:
Volume of a sphere: 4/3 π r³
4/3 (3.14) (2.5)³ =
4/3 (3.14) (15.625) = 65.42 units³
Volume of a cylinder = π r² h
(3.14) (2.5)² (4)
(3.14) (6.25)(4) = 78.5 units²
Volume of a Cone = 1/3 π r² h
(1/3)(3.14)(2.5)²(4) =
(1/3)(3.14)(6.25)(4) = 26.17 units²
Jack's bill at a restaurant came to $51.31 he wants to leave a 15% tip how much will the new total be including tip
The new total bill including the tip of 15% is $59.01.
Calculating the new bill including the tipTo find the amount of the tip, we can multiply the total bill by the percentage as a decimal:
tip = 0.15 * $51.31 = $7.70
To find the new total including the tip, we can add the tip to the original bill:
new total = $51.31 + $7.70 = $59.01
So the new total, including a 15% tip, will be $59.01.
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Andre is playing greatest product. he says the greatest product it’s possible to make in the game is 987x65 do you agree or disagree with andre
Andre's claim that the greatest product possible in the game is 987x65 is incorrect.
Why is Andre's claim incorrect?
While 987x65 is a large product, it is not the greatest possible product in the game. In fact, a larger product can be obtained by multiplying the two largest numbers available in the game. Without knowing the specific rules of the game, it is impossible to determine the exact greatest product, but it is certain that 987x65 is not it.
To elaborate, the game likely has certain constraints or rules that limit the numbers that can be multiplied. It may be possible to combine multiple numbers to create a larger product, or to find a different pair of numbers that yield a larger product. Therefore, without knowing the specifics of the game's rules, it is impossible to determine the greatest possible product.
The actual greatest product possible in the game will depend on the specific rules and constraints that are in place. It may be possible to combine multiple numbers to create an even larger product, or to find a different pair of numbers that yield a larger product. Without knowing the specifics of the game's rules, it is impossible to determine the greatest possible product.
In mathematics, the concept of maximum or greatest products is important and is studied in various areas such as algebra, number theory, and calculus. In real-world applications, maximum products play a critical role in determining profits, yields, and returns on investments in economics and finance. In engineering, the concept of maximum product is used in optimization problems, where the goal is to maximize or minimize a certain function or output.
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NEED HELP FAST!!!! Please answer both questions
Therefore, the molarity of the sugar solution is 0.3704 M at 25°C. Therefore, the molality of the NaCl solution is 1.8994 mol/kg.
What is equation?In mathematics, an equation is a statement that asserts the equality of two expressions. An equation typically consists of two sides separated by an equal sign (=). The expressions on either side of the equal sign may contain variables, constants, coefficients, and mathematical operations.
Here,
1. To calculate the molarity of a sugar solution, we need to first determine the number of moles of solute (glucose, C6H12O6) present in the solution. We can then divide this number of moles by the volume of the solution in liters to obtain the molarity. The number of moles of glucose in the solution can be calculated as follows:
Number of moles = mass of solute / molar mass of solute
Number of moles = 100.0 g / 180 g/mol
Number of moles = 0.5556 mol
Next, we can calculate the molarity of the solution using the following formula:
Molarity = number of moles / volume of solution (in L)
Molarity = 0.5556 mol / 1.50 L
Molarity = 0.3704 M
2. To calculate the molality of a solution, we need to know the number of moles of solute (NaCl) per kilogram of solvent (water).
First, let's calculate the number of moles of NaCl:
Number of moles = mass of NaCl / molar mass of NaCl
Number of moles = 200.0 g / 58.5 g/mol
Number of moles = 3.4188 mol
Next, we need to calculate the mass of the solvent (water) in kilograms:
Mass of solvent = 2.00 kg - 0.200 kg
Mass of solvent = 1.80 kg
Note that we subtracted the mass of the NaCl from the total mass of the solution to obtain the mass of the solvent.
Finally, we can calculate the molality of the solution using the following formula:
Molality = number of moles of solute / mass of solvent (in kg)
Molality = 3.4188 mol / 1.80 kg
Molality = 1.8994 mol/kg
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The angle of depression from the top of a 150m high cliff to a boat at sea is 7°. How much closer to the cliff must the boat move for the angle of depression to become 19°?
The boat must move 785.82 m closer to the cliff for the angle of depression to become 19°.
We need to find how much closer to the cliff the boat must move for the angle of depression to change from 7° to 19°.
Calculate the distance from the boat to the base of the cliff at 7° angle of depression.
Using the tangent function, we have:
tan(angle) = height/distance
tan(7°) = 150m/distance
distance = 150m/tan(7°)
distance=1221.49
Calculate the distance from the boat to the base of the cliff at 19° angle of depression.
Using the tangent function, we have:
tan(angle) = height/distance
tan(19°) = 150m/distance
distance = 150m/tan(19°)
distance=435.6665
Calculate the difference between the two distances to find out how much closer the boat must move.
difference = distance at 7° angle of depression - distance at 19° angle of depression
Plugging in the values from Steps 1 and 2, we get:
difference = (150m/tan(7°)) - (150m/tan(19°))
difference=1221.49-435.6665
difference=785.8235
After calculating, we find that the boat must move approximately 785.82 meters closer to the cliff for the angle of depression to change from 7° to 19°.
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If a data set has many very large data values, then the mean wil be
the median
a larger than
b. the same as
c. equal to
d. smaller than
A data set has many very large data values, then the mean will be larger than the median. The correct option is a.
In a dataset with large values, the mean is influenced by outliers and extreme values, whereas the median is not. The median is the value that divides the data into two equal halves, while the mean is calculated by summing up all values and dividing by the total number of values.
Thus, in a skewed dataset with large values, the mean tends to be pulled towards the larger values, resulting in a larger mean than the median. This effect is more pronounced in datasets with a high variance. Therefore, option (a) is the correct answer.
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Write a number that is greater than 10,910,099,999 but less than 11,000,000,000
Answer: 10, 911,000,000
Step-by-step explanation:
10,911,000,000 < 11,000,000,000
10,911,000,000 > 10,910,099,999
Write an equation and solve to find the numbers and show work.
9. Doubling the difference between a number and 4 is 6 more than the number.
10. Five less than the opposite of a number is three times the sum of the number and -7.
9. Let's start by defining the variable.
Let's say the number we're trying to find is "x".
Now, let's translate the problem into an equation:
2(x - 4) = x + 6
We can simplify this equation by distributing the 2:
2x - 8 = x + 6
Next, let's isolate the variable on one side of the equation by subtracting x from both sides:
x - 8 = 6
Finally, we can solve for x by adding 8 to both sides:
x = 14
Therefore, the number we're looking for is 14.
10. Let's start by defining the variable.
Let's say the number we're trying to find is "x".
Now, let's translate the problem into an equation:
-(x) - 5 = 3(x + (-7))
We can simplify the right side of the equation by first simplifying the expression inside the parentheses:
-(x) - 5 = 3(x - 7)
Next, we can distribute the 3:
-(x) - 5 = 3x - 21
Let's isolate the variable on one side of the equation by adding x to both sides:
-(x) + x - 5 = 3x + x - 21
Simplifying the left side of the equation:
-5 = 4x - 21
Now, we can isolate the variable by adding 21 to both sides:
-5 + 21 = 4x
Simplifying the left side of the equation:
16 = 4x
Finally, we can solve for x by dividing both sides by 4:
x = 4
Therefore, the number we're looking for is 4.
1. A curve C in R3 is given by x = {3 – 1, y = -3t2 + 2, z = 8t – 2. Find parametric equations for the tangent line to C at P = (0, -1,6).
The parametric equations for the tangent line to curve C at point P are: x = -t y = 6t - 1 z = 8t + 6
To find the parametric equations for the tangent line to curve C at point P, we need to first find the derivative of the curve at P.
Taking the derivative of each component of the curve, we get:
dx/dt = -1
dy/dt = -6t
dz/dt = 8
At point P = (0, -1, 6), t = -1.
Plugging this into the derivative, we get:
dx/dt = -1
dy/dt = 6
dz/dt = 8
So, the tangent vector to curve C at point P is < -1, 6, 8 >.
To get the parametric equations for the tangent line, we can use the point-slope form: r(t) = P + t< -1, 6, 8 > Plugging in the coordinates of point P, we get: r(t) = <0, -1, 6> + t< -1, 6, 8 > Expanding this out, we get: r(t) = <-t, 6t - 1, 8t + 6>
So, the parametric equations for the tangent line to curve C at point P are: x = -t y = 6t - 1 z = 8t + 6
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Find the critical points of f(x) = x - 18x² + 96x and use the Second Derivative Test (if possible) to determine whether each corresponds to a local minimum or maximum. (Use symbolic notation and fractions when needed)
To find the critical points of f(x) = x - 18x² + 96x, we need to find the values of x where f'(x) = 0.
f'(x) = 1 - 36x + 96
Setting f'(x) = 0, we get:
-36x + 97 = 0
x = 97/36
So the critical point is (97/36, f(97/36)).
To use the Second Derivative Test, we need to find f''(x):
f''(x) = -36
At the critical point x = 97/36, f''(97/36) = -36 < 0.
Since f''(97/36) is negative, the Second Derivative Test tells us that the critical point corresponds to a local maximum.
Therefore, the critical point (97/36, f(97/36)) is a local maximum.
To find the critical points of the function f(x) = x - 18x² + 96x, we first need to find its first derivative, f'(x), and then set it to zero to find the critical points.
1. Find the first derivative, f'(x):
f'(x) = d/dx (x - 18x² + 96x) = 1 - 36x + 96
2. Set f'(x) to zero and solve for x:
0 = 1 - 36x + 96
36x = 95
x = 95/36
Now, let's use the Second Derivative Test to determine if this critical point corresponds to a local minimum or maximum.
3. Find the second derivative, f''(x):
f''(x) = d/dx (1 - 36x + 96) = -36
4. Evaluate f''(x) at the critical point x = 95/36:
f''(95/36) = -36
Since f''(95/36) is negative, the Second Derivative Test tells us that the critical point x = 95/36 corresponds to a local maximum.
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In one month 382 adults and 65 children stayed in a hotel. How many people are there altogether?
In one month, a total of 447 people stayed at the hotel.
In one month, a hotel had 382 adults and 65 children staying as guests.
To find out the total number of people who stayed at the hotel, we simply need to add the number of adults and children together.
In one month, a total of 447 people (382 adults and 65 children) stayed at the hotel.
Overall, this problem is a simple example of addition in action. By adding the number of adults and children together, we can determine the total number of people who stayed in the hotel.
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